DIELECTRIC-COATED WAVEGUIDE 



1261 



in which 

 -£"1,2(2) = wave amplitudes in mode 1 (here always TEoi) and mode 2 

 (TMii or one of the TEi„), respectively; 

 7i,2 = propagation constant of mode 1 and 2, respectively (the 

 small perturbation of 71,2 caused by the coupling may be 

 neglected here) ; and 

 c = coupling coefficient between modes 1 and 2. 

 Subject to the initial conditions: 



EM = 1, E-,(0) = 0, 



the solution of (15) is: 



1 



1 + 



A7 



VAt^ - 4c2 



.-■ + 2 11 



^' 



A7 



^/Ay'^ — 4c2 



] 



0-^22 



Eo = 



jc 



(16) 



VA72 - 4c2 



[e 



-Tz^ 



-Tiz 



I 



where A7 = 71 — 72 and ri,2 = 3^[ti + 72 ± \/A7- — 4c2]. Ti and r2 

 are the propagation constants of the two coupled line normal modes. 

 Both coupled line normal modes are excited by the initial conditions. 

 For I C/A7 I « 1, ec^uations (16) can be simplified: 



E, = 



1 - 2-%sinh^A72e'^'' 

 A7- 2 



-(71 — Cc2/A7))2 



E, = /^sinhJA7^e~^^"^+'^^^ 

 A7 2 



(17) 



We are concerned with a difference in phase constant which is much 

 larger than the difference in attenuation constant. Consequently in 



A7 = jAjS + Aa we have | A,8 | » | Aa | and we may write : 



El = 



l+.Jsin^A^.-- 



•exp 



^\^^+A^^^^>~ 



r'A^^'^V' 



1 (18) 



E2 = i ^ -sin-AiSs exp - j- (pi + 13-^z --(«!+ a^^z 



We note that the amplitude A'l , apart from suffering an attenuation, 

 varies in an oscillatory manner, the maximum being ^1 = 1 and the 

 minimum £"1 = 1 — 2(cVA/5"). Accordingly^, the maximum mode conver- 

 sion loss is given b}^ : 



